Solving For 'x': Perimeter Of A Rectangle

Hey guys! Let's dive into a classic geometry problem. We're going to figure out the value of 'x' in a rectangular plot of land. The cool part? We've got the perimeter, which is 128 meters, and the dimensions are given in terms of 'x'. This is a great way to flex those math muscles and understand how algebra works in the real world. Buckle up; it's going to be fun!

Understanding the Problem: Unveiling the Rectangle's Secrets

So, the main goal here is to find the value of 'x'. We know that the plot of land is a rectangle. Remember those shapes from elementary school? A rectangle has four sides, and opposite sides are equal in length. We're given the perimeter, which is the total distance around the rectangle. Think of it like putting a fence around the land; the perimeter is the length of the fence. The dimensions are the key to solving this puzzle. One side is 2x, and the other is 2(x + 4). Let's break this down further, shall we?

First, what exactly is perimeter? Perimeter is the total length of all the sides added together. For a rectangle, we have two lengths and two widths. Given that the opposite sides of a rectangle are equal, the formula for the perimeter (P) of a rectangle is: P = 2 * (length + width). Now, in our specific problem, we know the perimeter (P = 128 meters), and we're given the dimensions in terms of 'x'. We have one side is 2x which we can define as the length. The other side is 2(x + 4), which can be defined as the width. So, now we can construct an equation. This is like building a bridge between the information we have and the answer we need. The equation will help us find the value of x. Let’s get to it! This is not as hard as it sounds. We are going to take this step by step, with all the pieces of information we already know. Keep in mind that the most important thing is the formula. We have the perimeter (P), the formula for the perimeter of a rectangle, and the sides. With all this information, it’s just a matter of plugging and chugging.

In our case, with dimensions of 2x and 2(x+4) and perimeter of 128 meters, we're set to find the unknown variable, 'x'. The initial step in tackling this is to substitute the given values into the perimeter formula. We'll replace the 'length' with '2x', the 'width' with '2(x + 4)', and 'P' with '128'. From here, we can simplify the equation and solve for 'x'. By doing so, we're able to determine the actual dimensions of the rectangle. What we're doing right now is translating a geometric problem into an algebraic one, where 'x' is that missing puzzle piece. It's all about making connections and using what we know to uncover what we don't. This process teaches us to think logically and systematically. It's like following clues to find a treasure. The clues are given to us by the equation.

Setting Up the Equation: The Foundation for Success

Alright, let's put our thinking caps on and get down to business. Remember the perimeter formula? It’s P = 2 * (length + width). We know P = 128, length = 2x, and width = 2(x + 4). Now, let's plug those values into the formula. So we end up with: 128 = 2 * (2x + 2(x + 4)).

Now, we're going to simplify things. We'll distribute the 2 on the right side of the equation. This means multiplying both terms inside the parenthesis by 2. Be careful with the signs and the order of operations. This gives us: 128 = 2 * (2x + 2x + 8). Then, let's simplify further by combining like terms within the parenthesis: 128 = 2 * (4x + 8). At this step, remember that we have to multiply the two by both terms inside the parenthesis. This yields: 128 = 8x + 16. The equation is now ready for the next step, solving for 'x'. Keep in mind that we are trying to isolate 'x' to find its value. In the process, we are going to have to use the properties of equality.

We are nearly at the end. You are doing great. The most important thing is to get the equation and the order of operations right. By following these steps you can be sure that you'll arrive at the correct answer. Remember to take your time. Don’t rush. Double-check your work. Always be careful when calculating. The key is to work systematically and methodically.

Solving for 'x': The Grand Finale

Let's get to the final step and solve for 'x'! We have the equation 128 = 8x + 16. To isolate 'x', we need to get rid of the 16 that's being added to 8x. We'll do this by subtracting 16 from both sides of the equation. Remember, whatever we do to one side, we must do to the other side to keep the equation balanced. So, 128 - 16 = 8x + 16 - 16. Which simplifies to 112 = 8x. Nice! We are now one step away from finding the value of 'x'.

Now, to isolate 'x' completely, we need to get rid of the 8 that's multiplying it. We'll do this by dividing both sides of the equation by 8. So, 112 / 8 = 8x / 8. This simplifies to 14 = x. So, x = 14. Woah! We did it! We found the value of 'x'. Now, we know the value of x, we can find the dimensions of the rectangle. One side is 2x which is 2 * 14 = 28 meters. The other side is 2(x + 4) which is 2 * (14 + 4) = 2 * 18 = 36 meters. With these dimensions, we can calculate the perimeter. We can plug the values into the formula: 2 * (28 + 36) = 2 * 64 = 128. And there you have it! The perimeter does match the information given at the beginning. So we know our answer is correct. You've now successfully solved for 'x' and understood how it relates to the dimensions of a rectangle. Congrats, guys! You have learned something valuable. I hope you enjoyed it. Feel free to explore more problems like this and test your knowledge. Remember that with some practice, anyone can master these concepts.

Choosing the Correct Alternative

Let's take a look at the options. However, it looks like we made a mistake. We calculated x = 14, and the answer choices are, in fact: A) 24 meters B) 32 meters C) 40 meters D) 48 meters. The issue seems to be that the answer choices provided in the original problem are not asking for the value of 'x', but the dimensions of the rectangle. So, let's see what dimensions give the correct perimeter. Option A, 24 meters is not a possibility. The correct option should be x = 14. If the question was about the actual size of the sides, the correct answer must be that the sides measure 28 and 36. As shown previously, these dimensions give a perimeter of 128 meters, which is the perimeter specified in the prompt. Therefore, based on the data given, the answer choices are not correct.